Properties

Label 6032.2.a.ba
Level 6032
Weight 2
Character orbit 6032.a
Self dual yes
Analytic conductor 48.166
Analytic rank 0
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6032.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1657624992\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 21 x^{8} + 40 x^{7} + 138 x^{6} - 243 x^{5} - 318 x^{4} + 448 x^{3} + 312 x^{2} - 240 x - 128\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3016)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{9} q^{5} -\beta_{2} q^{7} + ( 1 + \beta_{2} - \beta_{7} - \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{9} q^{5} -\beta_{2} q^{7} + ( 1 + \beta_{2} - \beta_{7} - \beta_{9} ) q^{9} + ( -1 - \beta_{4} - \beta_{7} ) q^{11} + q^{13} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{8} ) q^{15} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{9} ) q^{17} + ( 1 - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{19} + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} - \beta_{9} ) q^{21} + ( -3 + \beta_{3} - \beta_{5} - \beta_{9} ) q^{23} + ( 3 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{8} ) q^{25} + ( -1 - \beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{27} - q^{29} + ( 1 - \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{31} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{33} + ( 3 - 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{35} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{37} -\beta_{1} q^{39} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{41} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{43} + ( 5 + \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{9} ) q^{45} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{8} - 2 \beta_{9} ) q^{47} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{9} ) q^{49} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{51} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} ) q^{53} + ( -\beta_{1} + 4 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{9} ) q^{55} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{57} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{59} + ( 2 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{61} + ( -2 - \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{63} -\beta_{9} q^{65} + ( 4 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{67} + ( -2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{8} - \beta_{9} ) q^{69} + ( 2 - 3 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{71} + ( 5 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{8} + \beta_{9} ) q^{73} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{75} + ( 6 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} + 4 \beta_{9} ) q^{77} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{9} ) q^{79} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{81} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{83} + ( 4 - \beta_{1} - \beta_{2} - 3 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{85} + \beta_{1} q^{87} + ( -3 + \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{89} -\beta_{2} q^{91} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{93} + ( -4 + \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{9} ) q^{95} + ( 3 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{97} + ( 3 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 2q^{3} + 5q^{5} + 2q^{7} + 16q^{9} + O(q^{10}) \) \( 10q - 2q^{3} + 5q^{5} + 2q^{7} + 16q^{9} - 4q^{11} + 10q^{13} - 8q^{15} + 10q^{17} + q^{19} + q^{21} - 23q^{23} + 25q^{25} - 2q^{27} - 10q^{29} + 13q^{31} + 15q^{33} + 12q^{35} - 7q^{37} - 2q^{39} + 16q^{41} + 12q^{43} + 55q^{45} - 11q^{47} + 25q^{51} + 11q^{53} - 22q^{55} - 6q^{57} + 11q^{59} + 34q^{61} - 37q^{63} + 5q^{65} + 23q^{67} + 2q^{69} + 4q^{71} + 39q^{73} - 11q^{75} + 32q^{77} - 5q^{79} + 38q^{81} - 6q^{83} + 45q^{85} + 2q^{87} - 24q^{89} + 2q^{91} + 13q^{93} - 33q^{95} + 19q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 21 x^{8} + 40 x^{7} + 138 x^{6} - 243 x^{5} - 318 x^{4} + 448 x^{3} + 312 x^{2} - 240 x - 128\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -309 \nu^{9} + 1962 \nu^{8} + 3405 \nu^{7} - 38696 \nu^{6} + 14730 \nu^{5} + 219863 \nu^{4} - 196674 \nu^{3} - 304604 \nu^{2} + 207928 \nu + 68544 \)\()/25808\)
\(\beta_{3}\)\(=\)\((\)\( -291 \nu^{9} + 1143 \nu^{8} + 5587 \nu^{7} - 23209 \nu^{6} - 29820 \nu^{5} + 142003 \nu^{4} + 34323 \nu^{3} - 261240 \nu^{2} - 17852 \nu + 120552 \)\()/12904\)
\(\beta_{4}\)\(=\)\((\)\( -1045 \nu^{9} + 3190 \nu^{8} + 20285 \nu^{7} - 60436 \nu^{6} - 112174 \nu^{5} + 325151 \nu^{4} + 141498 \nu^{3} - 411660 \nu^{2} + 2696 \nu + 49648 \)\()/25808\)
\(\beta_{5}\)\(=\)\((\)\( 929 \nu^{9} - 1138 \nu^{8} - 20009 \nu^{7} + 21208 \nu^{6} + 134742 \nu^{5} - 110859 \nu^{4} - 299958 \nu^{3} + 113564 \nu^{2} + 166232 \nu + 10912 \)\()/12904\)
\(\beta_{6}\)\(=\)\((\)\( 539 \nu^{9} - 1136 \nu^{8} - 10293 \nu^{7} + 21698 \nu^{6} + 54768 \nu^{5} - 119693 \nu^{4} - 64256 \nu^{3} + 162242 \nu^{2} + 12668 \nu - 44896 \)\()/6452\)
\(\beta_{7}\)\(=\)\((\)\( 2835 \nu^{9} - 2372 \nu^{8} - 62811 \nu^{7} + 41478 \nu^{6} + 446726 \nu^{5} - 189645 \nu^{4} - 1132320 \nu^{3} + 39100 \nu^{2} + 838608 \nu + 274656 \)\()/25808\)
\(\beta_{8}\)\(=\)\((\)\( 2845 \nu^{9} - 4440 \nu^{8} - 60165 \nu^{7} + 83238 \nu^{6} + 399394 \nu^{5} - 439723 \nu^{4} - 910972 \nu^{3} + 476836 \nu^{2} + 659408 \nu + 66976 \)\()/25808\)
\(\beta_{9}\)\(=\)\((\)\( -1572 \nu^{9} + 2167 \nu^{8} + 33108 \nu^{7} - 40087 \nu^{6} - 215998 \nu^{5} + 204754 \nu^{4} + 467823 \nu^{3} - 184756 \nu^{2} - 315340 \nu - 51440 \)\()/12904\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{9} - \beta_{7} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{5} + \beta_{3} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-12 \beta_{9} + \beta_{8} - 12 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 8 \beta_{2} - \beta_{1} + 30\)
\(\nu^{5}\)\(=\)\(12 \beta_{9} + 18 \beta_{8} - 16 \beta_{7} + 23 \beta_{5} + 2 \beta_{4} + 14 \beta_{3} + 2 \beta_{2} + 60 \beta_{1} + 12\)
\(\nu^{6}\)\(=\)\(-122 \beta_{9} + 22 \beta_{8} - 131 \beta_{7} - 32 \beta_{6} + 25 \beta_{5} + 25 \beta_{4} + 13 \beta_{3} + 60 \beta_{2} - 14 \beta_{1} + 278\)
\(\nu^{7}\)\(=\)\(127 \beta_{9} + 240 \beta_{8} - 209 \beta_{7} + 6 \beta_{6} + 243 \beta_{5} + 42 \beta_{4} + 172 \beta_{3} + 37 \beta_{2} + 556 \beta_{1} + 158\)
\(\nu^{8}\)\(=\)\(-1223 \beta_{9} + 327 \beta_{8} - 1414 \beta_{7} - 396 \beta_{6} + 395 \beta_{5} + 395 \beta_{4} + 158 \beta_{3} + 460 \beta_{2} - 135 \beta_{1} + 2777\)
\(\nu^{9}\)\(=\)\(1295 \beta_{9} + 2899 \beta_{8} - 2555 \beta_{7} + 136 \beta_{6} + 2590 \beta_{5} + 647 \beta_{4} + 2013 \beta_{3} + 533 \beta_{2} + 5389 \beta_{1} + 2120\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.35685
2.58509
2.22036
1.29348
1.09823
−0.478552
−0.866798
−1.30047
−2.71446
−3.19372
0 −3.35685 0 3.43029 0 0.483428 0 8.26842 0
1.2 0 −2.58509 0 1.19710 0 −3.50800 0 3.68268 0
1.3 0 −2.22036 0 −0.143149 0 −1.44357 0 1.92998 0
1.4 0 −1.29348 0 −3.79162 0 2.99662 0 −1.32691 0
1.5 0 −1.09823 0 4.16520 0 1.76426 0 −1.79389 0
1.6 0 0.478552 0 −1.65690 0 2.65348 0 −2.77099 0
1.7 0 0.866798 0 2.19814 0 4.35947 0 −2.24866 0
1.8 0 1.30047 0 −2.99209 0 −3.88835 0 −1.30878 0
1.9 0 2.71446 0 −0.957650 0 −1.45512 0 4.36832 0
1.10 0 3.19372 0 3.55069 0 0.0377760 0 7.19984 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(13\) \(-1\)
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6032.2.a.ba 10
4.b odd 2 1 3016.2.a.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3016.2.a.g 10 4.b odd 2 1
6032.2.a.ba 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):

\(T_{3}^{10} + \cdots\)
\(T_{5}^{10} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T + 9 T^{2} + 14 T^{3} + 39 T^{4} + 51 T^{5} + 114 T^{6} + 173 T^{7} + 384 T^{8} + 690 T^{9} + 1258 T^{10} + 2070 T^{11} + 3456 T^{12} + 4671 T^{13} + 9234 T^{14} + 12393 T^{15} + 28431 T^{16} + 30618 T^{17} + 59049 T^{18} + 39366 T^{19} + 59049 T^{20} \)
$5$ \( 1 - 5 T + 25 T^{2} - 86 T^{3} + 294 T^{4} - 817 T^{5} + 2327 T^{6} - 5790 T^{7} + 14895 T^{8} - 34302 T^{9} + 80756 T^{10} - 171510 T^{11} + 372375 T^{12} - 723750 T^{13} + 1454375 T^{14} - 2553125 T^{15} + 4593750 T^{16} - 6718750 T^{17} + 9765625 T^{18} - 9765625 T^{19} + 9765625 T^{20} \)
$7$ \( 1 - 2 T + 37 T^{2} - 66 T^{3} + 695 T^{4} - 1142 T^{5} + 8916 T^{6} - 13843 T^{7} + 87532 T^{8} - 127061 T^{9} + 684422 T^{10} - 889427 T^{11} + 4289068 T^{12} - 4748149 T^{13} + 21407316 T^{14} - 19193594 T^{15} + 81766055 T^{16} - 54353838 T^{17} + 213297637 T^{18} - 80707214 T^{19} + 282475249 T^{20} \)
$11$ \( 1 + 4 T + 58 T^{2} + 228 T^{3} + 1823 T^{4} + 6415 T^{5} + 39016 T^{6} + 122496 T^{7} + 620150 T^{8} + 1745777 T^{9} + 7680312 T^{10} + 19203547 T^{11} + 75038150 T^{12} + 163042176 T^{13} + 571233256 T^{14} + 1033142165 T^{15} + 3229555703 T^{16} + 4443074988 T^{17} + 12432815098 T^{18} + 9431790764 T^{19} + 25937424601 T^{20} \)
$13$ \( ( 1 - T )^{10} \)
$17$ \( 1 - 10 T + 84 T^{2} - 549 T^{3} + 3367 T^{4} - 17712 T^{5} + 92760 T^{6} - 433810 T^{7} + 2037792 T^{8} - 8843519 T^{9} + 38190552 T^{10} - 150339823 T^{11} + 588921888 T^{12} - 2131308530 T^{13} + 7747407960 T^{14} - 25148507184 T^{15} + 81271194823 T^{16} - 225275931477 T^{17} + 585963625044 T^{18} - 1185878764970 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 - T + 70 T^{2} + 121 T^{3} + 2330 T^{4} + 10851 T^{5} + 65598 T^{6} + 350344 T^{7} + 2011367 T^{8} + 6983411 T^{9} + 47284540 T^{10} + 132684809 T^{11} + 726103487 T^{12} + 2403009496 T^{13} + 8548796958 T^{14} + 26868150249 T^{15} + 109616902730 T^{16} + 108158480419 T^{17} + 1188849412870 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20} \)
$23$ \( 1 + 23 T + 417 T^{2} + 5304 T^{3} + 57639 T^{4} + 519480 T^{5} + 4144202 T^{6} + 28866362 T^{7} + 181047968 T^{8} + 1011554771 T^{9} + 5125703802 T^{10} + 23265759733 T^{11} + 95774375072 T^{12} + 351217026454 T^{13} + 1159717631882 T^{14} + 3343551461640 T^{15} + 8532640606071 T^{16} + 18059194170888 T^{17} + 32655680862177 T^{18} + 41426511213649 T^{19} + 41426511213649 T^{20} \)
$29$ \( ( 1 + T )^{10} \)
$31$ \( 1 - 13 T + 247 T^{2} - 2052 T^{3} + 22179 T^{4} - 123136 T^{5} + 953034 T^{6} - 3050062 T^{7} + 20650764 T^{8} - 12060901 T^{9} + 371095070 T^{10} - 373887931 T^{11} + 19845384204 T^{12} - 90864397042 T^{13} + 880146912714 T^{14} - 3525279137536 T^{15} + 19683944140899 T^{16} - 56455884155772 T^{17} + 210664086247927 T^{18} - 343715088088723 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 + 7 T + 302 T^{2} + 1786 T^{3} + 42842 T^{4} + 218890 T^{5} + 3772298 T^{6} + 16728532 T^{7} + 228184037 T^{8} + 874442425 T^{9} + 9904958640 T^{10} + 32354369725 T^{11} + 312383946653 T^{12} + 847350331396 T^{13} + 7069893791978 T^{14} + 15178698747730 T^{15} + 109920850814378 T^{16} + 169548332559538 T^{17} + 1060768795084142 T^{18} + 909732178565539 T^{19} + 4808584372417849 T^{20} \)
$41$ \( 1 - 16 T + 279 T^{2} - 3100 T^{3} + 35163 T^{4} - 314898 T^{5} + 2875146 T^{6} - 22257861 T^{7} + 173699748 T^{8} - 1181928433 T^{9} + 8081467998 T^{10} - 48459065753 T^{11} + 291989276388 T^{12} - 1534034037981 T^{13} + 8124475436106 T^{14} - 36482885982498 T^{15} + 167027915426283 T^{16} - 603738249031100 T^{17} + 2227794138924759 T^{18} - 5238110950303376 T^{19} + 13422659310152401 T^{20} \)
$43$ \( 1 - 12 T + 333 T^{2} - 3498 T^{3} + 52769 T^{4} - 473677 T^{5} + 5206712 T^{6} - 39947403 T^{7} + 355635782 T^{8} - 2351606712 T^{9} + 17738345590 T^{10} - 101119088616 T^{11} + 657570560918 T^{12} - 3176098170321 T^{13} + 17800712192312 T^{14} - 69634518254911 T^{15} + 333572006732681 T^{16} - 950821501652286 T^{17} + 3892170692441133 T^{18} - 6031111343242116 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 + 11 T + 327 T^{2} + 2890 T^{3} + 47467 T^{4} + 341894 T^{5} + 4127976 T^{6} + 24821600 T^{7} + 253527788 T^{8} + 1341995161 T^{9} + 12718097874 T^{10} + 63073772567 T^{11} + 560042883692 T^{12} + 2577052976800 T^{13} + 20143206055656 T^{14} + 78411681823258 T^{15} + 511657014021643 T^{16} + 1464140818138070 T^{17} + 7786290738395847 T^{18} + 12310435204130437 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 - 11 T + 194 T^{2} - 1264 T^{3} + 14938 T^{4} - 67452 T^{5} + 908730 T^{6} - 4184418 T^{7} + 65915997 T^{8} - 345098255 T^{9} + 4252769656 T^{10} - 18290207515 T^{11} + 185158035573 T^{12} - 622963598586 T^{13} + 7170316799130 T^{14} - 28208122393836 T^{15} + 331091226545002 T^{16} - 1484834880753968 T^{17} + 12078379939804034 T^{18} - 36297399509823463 T^{19} + 174887470365513049 T^{20} \)
$59$ \( 1 - 11 T + 318 T^{2} - 2108 T^{3} + 43376 T^{4} - 210326 T^{5} + 4315126 T^{6} - 17386472 T^{7} + 341068447 T^{8} - 1124843795 T^{9} + 21665360440 T^{10} - 66365783905 T^{11} + 1187259264007 T^{12} - 3570816232888 T^{13} + 52287939502486 T^{14} - 150367168111474 T^{15} + 1829622827212016 T^{16} - 5246077329998452 T^{17} + 46692079158174078 T^{18} - 95292954005204329 T^{19} + 511116753300641401 T^{20} \)
$61$ \( 1 - 34 T + 993 T^{2} - 20022 T^{3} + 350533 T^{4} - 5098624 T^{5} + 65689568 T^{6} - 743603065 T^{7} + 7549454292 T^{8} - 68794909067 T^{9} + 565491672106 T^{10} - 4196489453087 T^{11} + 28091519420532 T^{12} - 168783767296765 T^{13} + 909527313886688 T^{14} - 4306278970589824 T^{15} + 18059591385884413 T^{16} - 62923997062812462 T^{17} + 190365361806300033 T^{18} - 397600967156360794 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 - 23 T + 575 T^{2} - 8656 T^{3} + 130595 T^{4} - 1528328 T^{5} + 17880102 T^{6} - 176859206 T^{7} + 1761396564 T^{8} - 15269035151 T^{9} + 133601892406 T^{10} - 1023025355117 T^{11} + 7906909175796 T^{12} - 53192705374178 T^{13} + 360304098894342 T^{14} - 2063434004531096 T^{15} + 11813412419360555 T^{16} - 52461519655675888 T^{17} + 233488914595068575 T^{18} - 625750291114783781 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 4 T + 245 T^{2} + 152 T^{3} + 34795 T^{4} + 64931 T^{5} + 4200656 T^{6} + 10515485 T^{7} + 385517396 T^{8} + 1066502494 T^{9} + 30346697046 T^{10} + 75721677074 T^{11} + 1943393193236 T^{12} + 3763607751835 T^{13} + 106745730262736 T^{14} + 117150415989781 T^{15} + 4457249379031195 T^{16} + 1382458264075432 T^{17} + 158209615155211445 T^{18} - 183394002873796124 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 - 39 T + 1239 T^{2} - 27592 T^{3} + 527678 T^{4} - 8396105 T^{5} + 118316613 T^{6} - 1461221190 T^{7} + 16242913949 T^{8} - 161462180094 T^{9} + 1454773757536 T^{10} - 11786739146862 T^{11} + 86558488434221 T^{12} - 568439883670230 T^{13} + 3359983690277733 T^{14} - 17405726767345265 T^{15} + 79855741859726942 T^{16} - 304819819938924424 T^{17} + 999204053856766359 T^{18} - 2295991881622448607 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 + 5 T + 612 T^{2} + 3077 T^{3} + 175970 T^{4} + 855105 T^{5} + 31629008 T^{6} + 142880874 T^{7} + 3964821173 T^{8} + 16046018225 T^{9} + 363647827736 T^{10} + 1267635439775 T^{11} + 24744448940693 T^{12} + 70445843236086 T^{13} + 1231952423549648 T^{14} + 2631206312066895 T^{15} + 42776099548030370 T^{16} + 59090427950411243 T^{17} + 928470591662815332 T^{18} + 599257979913091595 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 + 6 T + 449 T^{2} + 3389 T^{3} + 99750 T^{4} + 882638 T^{5} + 15179264 T^{6} + 142095200 T^{7} + 1778859513 T^{8} + 15981810109 T^{9} + 165482723742 T^{10} + 1326490239047 T^{11} + 12254563185057 T^{12} + 81248188122400 T^{13} + 720382383455744 T^{14} + 3476746955056234 T^{15} + 32612302243557750 T^{16} + 91964076803845903 T^{17} + 1011279212230429409 T^{18} + 1121641531605242418 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( 1 + 24 T + 748 T^{2} + 13791 T^{3} + 264066 T^{4} + 3895127 T^{5} + 56933846 T^{6} + 697690076 T^{7} + 8353976589 T^{8} + 86608409258 T^{9} + 874367506268 T^{10} + 7708148423962 T^{11} + 66171848561469 T^{12} + 491849875187644 T^{13} + 3572157086788886 T^{14} + 21750620729405023 T^{15} + 131235861578907426 T^{16} + 609994339544240439 T^{17} + 2944568426665156588 T^{18} + 8408553688979645016 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 - 19 T + 652 T^{2} - 9479 T^{3} + 197022 T^{4} - 2353457 T^{5} + 37409138 T^{6} - 382428806 T^{7} + 5132486585 T^{8} - 46431033299 T^{9} + 553064521652 T^{10} - 4503810230003 T^{11} + 48291566278265 T^{12} - 349032445658438 T^{13} + 3311804089969778 T^{14} - 20209936039218449 T^{15} + 164113810355121438 T^{16} - 765886938568033127 T^{17} + 5110006703533778572 T^{18} - 14444390114436739123 T^{19} + 73742412689492826049 T^{20} \)
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